Two materials with the same thermal conductivity can behave completely differently when the temperature changes. One reaches a new steady state in seconds, the other takes hours. Conductivity does not explain that, because conductivity only tells you how much heat flows once nothing is changing any more. The property that governs the timescale is thermal diffusivity, written α, and it is the ratio of how fast a material moves heat to how much heat it has to store on the way.
Arb Digital builds free physics calculators that each own one job. This page owns α = k ÷ (ρc) and the transient quantities that follow from it. The live thermal conductivity calculator applies the steady plane-slab form of Fourier's law and returns k, heat flux and R-value from a measured heat flow — a steady-state page that explicitly does not compute diffusivity, density or specific heat. The live Biot number calculator decides whether a body can be treated as having one uniform temperature at all, which is the question you answer before choosing a transient model; this page supplies the diffusivity and the Fourier number you then use inside that model.
What This Thermal Diffusivity Calculator Does
Enter the thermal conductivity, density and specific heat and the calculator returns the diffusivity in square metres per second and in the square millimetres per second that tables actually use. It also returns the volumetric heat capacity ρc, which is the denominator on its own and is worth seeing separately, because it is the term that varies least between solids and most between a solid and a gas.
Two transient figures follow. The diffusion time over your characteristic length, L² ÷ α, is the order-of-magnitude time for a temperature change to make itself felt right across that distance. The thermal penetration depth, √(αt), is the complementary question: how far a change at a surface has reached after a given time. The note beneath the results also reports the Fourier number, αt ÷ L², which is dimensionless time and the parameter every transient conduction chart is plotted against.
The reverse mode solves for conductivity instead. Laser-flash and hot-disk instruments measure diffusivity directly rather than conductivity, and the published route back to k is to multiply the measured α by a separately measured density and specific heat. Doing it in that order matters, because the uncertainties in ρ and c then carry into k rather than the other way round.
How to Use It
- Enter your own material properties. This page deliberately publishes no material table, because k, ρ and c all vary with temperature, alloy, moisture content and porosity, and a generic figure is often wrong by a factor that matters.
- Check the units. Specific heat must be per kilogram, not per gram or per mole, and density in kilograms per cubic metre. A specific heat entered in J/g·K makes the diffusivity a thousand times too small.
- Set a characteristic length that reflects the distance heat actually has to travel — half the thickness for a slab heated on both sides, the full thickness for one side.
- Set the elapsed time to whatever your process actually takes, and compare the penetration depth against the part's size.
- Switch to reverse mode if you have a measured diffusivity and need the conductivity that goes with it.
The Formula: Why Alpha Is a Ratio
Thermal diffusivity is defined as
α = k ÷ (ρ × c), with units of m²/s.
The numerator is the ability to transport heat. The denominator, ρc, is the volumetric heat capacity: the energy it takes to raise one cubic metre of the material by one kelvin. Diffusivity is transport divided by storage, and that is exactly why it has units of area per unit time rather than anything that looks like a heat flow. It falls out of the transient conduction equation, ∂T/∂t = α∇²T, where it is the only material property that appears at all — the shape of a transient temperature field depends on α and on nothing else about the material.
The steady conduction relation that supplies k is the one OpenStax gives in section 1.6, Mechanisms of Heat Transfer, of University Physics Volume 2, where the rate of conduction through a slab is P = kA(Th − Tc) ÷ d. Georgia State's HyperPhysics pages on heat transfer set out the same conduction relation alongside convection and radiation.
Work the default by hand. Copper with k = 401 W/m·K, ρ = 8,960 kg/m³ and c = 385 J/kg·K has ρc = 8,960 × 385 = 3,449,600 J/m³·K. Then α = 401 ÷ 3,449,600 = 1.1625 × 10−4 m²/s, or 116.25 mm²/s. Over a 10 mm length the diffusion time is L² ÷ α = 0.0001 ÷ 0.00011625 = 0.86 seconds. After 60 seconds the penetration depth is √(0.00011625 × 60) = √0.006975 = 0.0835 m, or 83.5 mm — which is why a copper bar heated at one end gets uncomfortable to hold very quickly.
Why Conductivity and Diffusivity Rank Materials Differently
The two properties do not order materials the same way, and that is the most useful thing on this page.
Copper conducts heat about three times better than aluminium alloys of similar structure, but its density and volumetric heat capacity are higher too, so the diffusivity gap between them is much narrower than the conductivity gap. Stone and concrete conduct poorly and store enormously, giving a very low diffusivity, which is precisely what makes thermal mass work in a building: heat entering the wall during the day takes many hours to reach the inside. A material chosen to slow a temperature swing down needs low diffusivity, while a heat sink base needs high diffusivity so a transient hot spot spreads before it builds.
The gap is starkest in gases. Air has a very low conductivity but also a tiny volumetric heat capacity, and the two nearly cancel: air's diffusivity is comparable to that of many metals even though its conductivity is thousands of times lower. That is not a paradox. It means a temperature change propagates through still air at a respectable rate, while almost no energy travels with it — which is exactly why still air is a good insulator and a poor thermal buffer at the same time.
The Fourier Number and Dimensionless Time
The Fourier number, Fo = αt ÷ L², is the single most useful thing diffusivity gives you. It is dimensionless, and it is the horizontal axis of every transient conduction chart ever drawn.
Its value tells you where you are in the process without needing to know anything else. Below about 0.05 the temperature change has barely penetrated and the body behaves as if it were semi-infinite: only a thin skin near the surface knows anything has happened. Around 0.2 the change has reached the centre and the standard one-term approximations to the exact series solutions become accurate. Above roughly 1 the body is close to its new steady state, and by 2 or 3 the transient is essentially over.
This gives an immediate practical scaling. Because Fo depends on L², doubling the thickness of a part quadruples the time to heat it through. That is why quenching, tempering, curing and cooking times scale with the square of thickness rather than in proportion to it, and it is why a thin section reaches temperature so disproportionately faster than a thick one. The default 60 seconds and 10 mm above give a Fourier number of about 70, which is deep in the settled regime — that copper is long since uniform.
Where the Simple Ratio Stops Being Enough
All three inputs are temperature-dependent, and the composed ratio inherits every one of those dependencies. Conductivity of a pure metal falls with temperature while that of most ceramics and glasses rises; specific heat generally rises; density falls with thermal expansion. Quoting a single diffusivity for a process spanning hundreds of degrees hides real variation, and the honest approach is to evaluate the properties at a representative mean temperature and to check the sensitivity at both ends.
The relation also assumes a homogeneous, isotropic, non-moving solid with no internal heat generation and no phase change. Composites and laminates have direction-dependent conductivity and therefore direction-dependent diffusivity. Moist porous materials transport heat partly by moisture movement, which the ratio cannot see at all. And any latent heat — melting, freezing, curing, boiling — absorbs or releases energy at nearly constant temperature and stalls the transient completely, which is why the latent heat calculator exists as a separate page.
Finally, diffusivity governs conduction inside the body only. What happens at the surface is convection or radiation, and that boundary condition is at least as important as the interior property. The heat transfer coefficient calculator and the Newton's law of cooling calculator cover that side, the specific heat calculator handles Q = mcΔT for a single body, and the calorimetry calculator finds the final temperature when two bodies are mixed. Those are energy-balance questions; diffusivity is a rate-of-spreading question, and the two are answered with different tools.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Treating diffusivity as a stand-in for conductivity — they rank materials differently. High conductivity with high volumetric heat capacity gives a moderate diffusivity, and air has a diffusivity comparable to metals despite conducting almost nothing.
- Entering specific heat per gram — the formula needs J/kg·K. A value in J/g·K is out by a factor of a thousand and the diffusivity comes out a thousand times too small.
- Using room-temperature properties for a high-temperature process — all three inputs change with temperature, and the ratio inherits every one of those changes.
- Scaling heating time in proportion to thickness — the Fourier number depends on the square of the length, so doubling a section quadruples the time to heat it through.
- Applying it across a phase change — latent heat absorbs energy at nearly constant temperature and stalls the transient, and no single diffusivity describes that.
Related Free Tools From Arb Digital
For the steady-state side, the thermal conductivity calculator applies Fourier's law to a plane slab, and the thermal resistance calculator handles series and parallel conduction stacks. Before choosing a transient model, use the Biot number calculator to check whether a lumped-capacitance treatment is valid. The specific heat calculator gives Q = mcΔT for one body and the calorimetry calculator finds the equilibrium temperature when two are mixed. For surface losses, see the Newton's law of cooling calculator, the heat transfer coefficient calculator and the latent heat calculator. Everything Arb Digital publishes is on the free online tools hub.
Frequently Asked Questions
It is the ratio of a material's thermal conductivity to its volumetric heat capacity, written as alpha equals k divided by rho times c. It has units of square metres per second and measures how quickly a temperature change spreads through the material rather than how much heat flows.
Conductivity governs the steady heat flow through a material once nothing is changing. Diffusivity governs the speed at which a temperature change propagates, and it accounts for the energy the material must absorb along the way as well as the energy it transports.
The SI unit is square metres per second, but published values are almost always given in square millimetres per second because the SI figures are inconveniently small. One square millimetre per second equals one millionth of a square metre per second.
It is dimensionless time in transient conduction, equal to the diffusivity multiplied by the elapsed time and divided by the square of the characteristic length. Values below about 0.05 mean the change has barely penetrated, while values above about 1 mean the body is close to its new steady state.
Multiply the diffusivity by the density and by the specific heat capacity. Laser-flash and hot-disk instruments measure diffusivity directly, so this is the standard route back to conductivity, and it requires separate measurements of density and specific heat.
Because diffusivity is a ratio. Air conducts heat very poorly, but it also stores very little energy per unit volume, and the two effects nearly cancel. A temperature change therefore propagates through still air at a respectable rate even though almost no energy travels with it.
As an order of magnitude, the time is the distance squared divided by the diffusivity. The dependence on the square of the distance is why doubling the thickness of a part quadruples the time needed to heat it through rather than merely doubling it.
It is the square root of the diffusivity multiplied by the elapsed time, and it estimates how far a temperature change applied at a surface has reached after that time. It is the natural length scale for a body thick enough to be treated as semi-infinite.
This tool is provided for educational and study use. It evaluates a single defining ratio from properties you supply and assumes a homogeneous, isotropic solid with constant properties, no internal heat generation, no moisture transport and no phase change. Real materials have temperature-dependent and often direction-dependent properties, and any design of a load-bearing, safety-critical or thermally critical component should be reviewed by a qualified engineer against measured material data.
